Let $z_o$ be a fixed complex number and $z$ be a variable complex number
I've been told that the locus of $z$ in $|z-z_o|=r$ where $r\gt0$ is a circle centred at $z_o$
I have not been able to understand this . $z-z_o$ is a complex number originating at the origin so shouldn't the locus be centred at the origin ??
Convince yourself by studying the real case $|x-5|=1$ for instance.
Similarly $x-5$ is a real originating at the origin, yet $x=4\text{ or }6$ both points at distance $1$ of center $5$.
The complex case is just the two dimensional extension of the real case.